A061057 -id:A061057 - cp's OEIS frontend

A060777 Larger central (or median) divisor of n!.

1, 2, 3, 6, 12, 30, 72, 210, 630, 1920, 6336, 22176, 78975, 295680, 1144000, 4576000, 18869760, 80061696, 348986880, 1560176640, 7148445696, 33530112000, 160813154304, 787718131200, 3938590656000, 20083261440000, 104351051284480, 552173794099200, 2973519499493376, 16286922357866496, 90680032493568000, 512971179263262720

Offset: 1

Labos Elemer, Apr 26 2001

nonn

Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of y. Inequality "x < y" gives the same sequence, except that a(1) is not defined.

The integer part of square root of n! (A055226(n)) is situated between x and y.

			Divisors of 6!=720 are {1, 2, 3, 4, 5, 6, ..., 24, 30, ..., 360, 720}. a(6)=30, the 16th one from the 30 divisors of 720.

Max Alekseyev, Table of n, a(n) for n = 1..140
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A027423, A055226, A000196, A000142, A000005, A060776, A061057, A033677.

Table[ Part[ Divisors[ w! ], 1+Floor[ DivisorSigma[ 0, n! ]/2 ] ], {w, a, b} ]

a(n) = A033677(A000142(n)). - Pontus von Brömssen, Jul 15 2023

Sum_{k=1..n} a(k) = sqrt(n!) * (1 + O(1/n^c)), where c < 1 is a positive constant (De Koninck and Verreault, 2024, p. 48, Theorem 2.1). - Amiram Eldar, Dec 10 2024

More terms from Don Reble, Dec 13 2001

A060796 Upper central divisor of n-th primorial.

Original entry on oeis.org

2, 3, 6, 15, 55, 182, 715, 3135, 15015, 81345, 448630, 2733549, 17490603, 114388729, 785147363, 5708795638, 43850489690, 342503171205, 2803419704514, 23622001517543, 201817933409378, 1793779635410490, 16342166369958702, 154171363634898185, 1518410187442699518, 15259831781575946565

Offset: 1

Labos Elemer, Apr 27 2001

nonn

Also: Write product of first n primes as x*y with x < y and x maximal; sequence gives value of y. Indeed, p(n)# = primorial(n) = A002110(n) is never a square for n >= 1; all exponents in the prime factorization are 1. Therefore primorial(n) has N = 2^n distinct divisors. Since this is an even number, the N divisors can be grouped in N/2 pairs {d(k), d(N+1-k)} with product equal to p(n)#. One of the two is always smaller and one is larger than sqrt(p(n)#). This sequence gives the (2^(n-1)+1)-th divisor, which is the smallest one larger than sqrt(p(n)#). - M. F. Hasler, Sep 20 2011

			n = 8, q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3135, and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _M. F. Hasler_, Sep 20 2011]

Max Alekseyev, Table of n, a(n) for n = 1..70

Cf. A060755, A000196, A002110, A033677, A060776, A060777, A061057, A060795 (x), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A200744.

k = 1; Do[k *= Prime[n]; l = Divisors[k]; x = Length[l]; Print[l[[x/2 + 1]]], {n, 1, 24}] (* Ryan Propper, Jul 25 2005 *)

A060796(n) = divisors(prod(k=1,n,prime(k)))[2^(n-1)+1] \\ Requires stack size > 2^(n+5). - M. F. Hasler, Sep 20 2011

a(n) = A033677(A002110(n)).

a(n) = A002110(n)/A060795(n). - M. F. Hasler, Mar 21 2022

More terms from Ryan Propper, Jul 25 2005
a(24)-a(37) in b-file calculated from A182987 by M. F. Hasler, Sep 20 2011
a(38) from David A. Corneth, Mar 21 2022
a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A060795 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 2, 5, 14, 42, 165, 714, 3094, 14858, 79534, 447051, 2714690, 17395070, 114371070, 783152070, 5708587335, 43848093003, 342444658094, 2803119896185, 23619540863730, 201813981102615, 1793779293633437, 16342050964565645, 154170926013430326, 1518409177581024365

Offset: 1

Labos Elemer, Apr 27 2001

nonn

Or, lower central divisor of n-th primorial.

Subsequence of A005117 (squarefree numbers). - Michel Marcus, Feb 22 2016

			n = 8: q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3094 and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _Colin Barker_, Oct 22 2010]
2*3*5*7 = 210 = 14*15 with difference of 1, so a(4) = 14.

Max Alekseyev, Table of n, a(n) for n = 1..70

Cf. A000196, A060776, A060777, A061057, A060796 (y), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A060755, A033677, A200743.

F:= proc(n) local P,N,M;
     P:= {seq(ithprime(i),i=1..n)};
     N:= floor(sqrt(convert(P,`*`)));
     M:= map(convert, combinat:-powerset(P),`*`);
     max(select(`<=`,M,N))
end proc:
map(F, [$1..20]); # Robert Israel, Feb 22 2016

a[n_] := (m = Times @@ Prime[Range[n]] ; dd = Divisors[m]; dd[[Length[dd]/2 // Floor]]); Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Oct 15 2016 *)

a(n) = my(m=prod(i=1, n, prime(i))); divisors(m)[numdiv(m)\2]; \\ Michel Marcus, Feb 22 2016

a(n) = A060775(A002110(n)). - Labos Elemer, Apr 27 2001

a(n) = A002110(n)/A060796(n). - M. F. Hasler, Mar 21 2022

More terms from Ed Pegg Jr, May 28 2001
a(16)-a(23) computed by Jud McCranie, Apr 15 2000
a(24) and a(25) from Robert Israel, Feb 22 2016
a(25) corrected by Jean-François Alcover, Oct 15 2016
a(26)-a(33) in b-file from Amiram Eldar, Apr 09 2020
Up to a(38) using b-file of A060796, by M. F. Hasler, Mar 21 2022
a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A061060 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 1, 1, 1, 13, 17, 1, 41, 157, 1811, 1579, 18859, 95533, 17659, 1995293, 208303, 2396687, 58513111, 299808329, 2460653813, 3952306763, 341777053, 115405393057, 437621467859, 1009861675153, 6660853109087, 29075165225531

Offset: 1

Ed Pegg Jr, May 28 2001

nonn

			a(4)=1: 2*3*5*7 = 210 = 14*15, so we can take x=14, y=15, with difference of 1.
Also: n=3: 2*3-5=1; n=4: 3*5-2*7=1; n=5: 5*11-2*3*7=13; n=6: 2*7*13-3*5*11=17; n=7: 5*11*13-2*3*7*17=1; n=8: 3*5*11*19-2*7*13*17=41

Max Alekseyev, Table of n, a(n) for n = 1..70
C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009). - _Jonathan Sondow_, Jun 21 2012
Carlos Rivera, Conjecture 18. Minimal Primorial Partitions, The Prime Puzzles & Problems Connection.

Cf. A038667, A060776, A060777, A061057, A060795, A060796, A061030-A061033, A182987, A263292, A352813.

A061060aux := proc(l1,l2) local resul ; resul := product(l1[i],i=1..nops(l1)) ; resul := resul-product(l2[i],i=1..nops(l2)) ; RETURN(abs(resul)) ; end:
A061060 := proc(n) local plist,i,subl,resul,j,l1,l2,k,d ; plist := [] ; resul := 1 ; for i from 1 to n do resul := resul*ithprime(i) ; plist := [op(plist), ithprime(i)] ; od; for i from 1 to n/2 do subl := combinat[choose](plist,i) ; for j from 1 to nops(subl) do l1 := op(j,subl) ; l2 := convert(plist,set) minus convert(l1,set) ; d := A061060aux(l1,l2) ; if d < resul then resul := d ; fi ; od; od ; RETURN(resul) ; end:
for n from 3 to 19 do printf("%d,",A061060(n)) ; od ; # R. J. Mathar, Aug 26 2006 [This Maple program was attached to A121315. However I think it belongs here, so I renamed the variables and moved it to this entry. - N. J. A. Sloane, Sep 16 2005]

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{arrayofnprimes = Array[Prime, n], primorial = Times @@ Array[Prime, n], diffmin = Infinity, adiff, sub}, If[n == 1, 1, Do[sub = Times @@ NthSubset[i, arrayofnprimes]; adiff = Abs[primorial/sub - sub]; If[adiff < diffmin, diffmin = adiff], {i, 2, 2^n/2}]; diffmin]]; Do[ Print@f@n, {n, 30}] (* Robert G. Wilson v, Sep 14 2006 *)

Conjecture: Limit_{N->oo} (Sum_{n=1..N} log(a(n))) / (Sum_{n=1..N} prime(n)) = 1/e (A068985). - Alain Rocchelli, Nov 13 2023

Terms a(16)-a(45) in b-file computed by Jud McCranie, Apr 15 2000; Jan 12 2016
a(46)-a(60) in b-file from Don Reble, Jul 11 2020
a(61)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A038667 Minimal value of |product(A) - product(B)| where A and B are a partition of {1,2,3,...,n}.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 6, 2, 18, 54, 30, 36, 576, 576, 840, 6120, 24480, 20160, 93696, 420480, 800640, 1305696, 7983360, 80313120, 65318400, 326592000, 2286926400, 3002360256, 13680979200, 37744574400, 797369149440, 1763653953600, 16753029012720, 16880461678080, 10176199188480, 26657309952000

Offset: 0

Erich Friedman

nonn

Conjecture: The sequence of rational numbers A061057(n)/a(n) has 1 as a limit point. Question: What other limit points does the sequence have? - Richard Peterson, Jul 13 2023

			For n=1, we put 1 in one set and the other is empty; with the standard convention for empty products, both products are 1.
For n=13, the central pair of divisors of n! are 78975 and 78848. Since neither is divisible by 10, these values cannot be obtained. The next pair of divisors are 79200 = 12*11*10*6*5*2*1 and 78624 = 13*9*8*7*4*3, so a(13) = 79200 - 78624 = 576.

Max Alekseyev, Table of n, a(n) for n = 0..140 (terms for n = 0..60 from Peter J. Taylor)

Cf. A061033, A061057, A061060, A200743, A200744, A127180, A263292.

a:= proc(n) local l, ll, g, gg, p, i; l:= [i$i=1..n]; ll:= [i!$i=1..n]; g:= proc(m, j, b) local mm, bb, k; if j=1 then m else mm:= m; bb:= b; for k to 2 while (mmbb then bb:= max(bb, g(mm, j-1, bb)) fi; mm:= mm*l[j] od; bb fi end; Digits:= 700; p:= ceil(sqrt(ll[n])); gg:= g(1, nops(l), 1); ll[n]/gg -gg end: a(0):=0:
seq(a(n), n=0..20); #  Alois P. Heinz, Jul 09 2009, revised Oct 17 2015

a[n_] := Module[{l, ll, g, gg, p, i}, l = Range[n]; ll = Array[Factorial, n]; g[m_, j_, b_] := g[m, j, b] = Module[{mm, bb, k}, If[j==1, m, mm=m; bb=b; For[k=1, mm bb , bb = Max[bb, g[mm, j-1, bb]]]; mm = mm*l[[j]]]; bb]]; p = Ceiling[Sqrt[ ll[[n]]]]; gg = g[1, Length[l], 1]; ll[[n]]/gg - gg]; a[0]=0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 35}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz *)

from math import prod, factorial
from itertools import combinations
def A038667(n):
    m = factorial(n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for l in range(n,n//2,-1) for d in combinations(range(1,n+1),l)) # Chai Wah Wu, Apr 06 2022

a(n) = A200744(n) - A200743(n) = (A200744(n)^2 - A200743(n)^2) / A127180(n). - Max Alekseyev, Apr 08 2022

a(n) >= A061057(n).

a(28)-a(31) from Alois P. Heinz, Jul 09 2009
a(1) and examples from Franklin T. Adams-Watters, Nov 22 2011
a(32)-a(33) from Alois P. Heinz, Nov 23 2011
a(34)-a(35) from Alois P. Heinz, Oct 17 2015

A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n.

Original entry on oeis.org

0, 1, 2, 6, 18, 30, 576, 840, 24480, 93696, 800640, 7983360, 65318400, 2286926400, 13680979200, 797369149440, 16753029012720, 10176199188480, 159943859712000, 26453863460044800, 470500040794291200, 20720967220237197312, 61690805562507264000

Offset: 0

Peter J. Taylor, Apr 04 2022

nonn

a(n) >= A038667(2*n).
Conjecture: a(n) = A038667(2*n) for all n. It is verified for n<=70. - Max Alekseyev, Jun 18 2022
Bernardo Recamán Santos proposes that this should be called Luciana's sequence for the student whose question prompted its investigation. (See MathOverflow link below.)

			For n = 4, the partition A = {1,5,6,7} and B = {2,3,4,8} is optimal, giving difference 1*5*6*7 - 2*3*4*8 = 18.
_Rob Pratt_ computed the optimal solutions for n <= 10:
[ n]    a(n)                   partitions of 2n
------------------------------------------------------------------
[ 1]       1                         2 | 1
[ 2]       2                       2,3 | 1,4
[ 3]       6                     1,5,6 | 2,3,4
[ 4]      18                   1,5,6,7 | 2,3,4,8
[ 5]      30                2,3,4,8,10 | 1,5,6,7,9
[ 6]     576              1,4,7,8,9,11 | 2,3,5,6,10,12
[ 7]     840           2,4,5,6,8,11,14 | 1,3,7,9,10,12,13
[ 8]   24480        1,5,6,7,8,13,14,15 | 2,3,4,9,10,11,12,16
[ 9]   93696     2,3,6,8,9,11,12,13,18 | 1,4,5,7,10,14,15,16,17
[10]  800640  2,3,4,8,9,11,12,18,19,20 | 1,5,6,7,10,13,14,15,16,17

Max Alekseyev, Table of n, a(n) for n = 0..70
Gordon Hamilton, Thirsty Fractions, MathPickle, 2013, for elementary and middle school teachers.
MathOverflow discussion Splitting the integers from 1 to 2n into two sets with products as close as possible.

Cf. A061057, A038667.

from math import prod, factorial
from itertools import combinations
def A352813(n):
    m = factorial(2*n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for d in combinations(range(2,2*n+1),n-1)) # Chai Wah Wu, Apr 06 2022

def A352813(n):
    return min(abs(prod(A)-prod(B)) for (A,B) in SetPartitions((1..2*n), [n,n]))
[A352813(n) for n in (1..10)] # Freddy Barrera, Apr 05 2022

A323728 a(n) is the smallest number k such that both k-2n and k+2n are squares.

Original entry on oeis.org

2, 5, 10, 8, 26, 13, 50, 20, 18, 29, 122, 25, 170, 53, 34, 32, 290, 45, 362, 41, 58, 125, 530, 52, 50, 173, 90, 65, 842, 61, 962, 80, 130, 293, 74, 72, 1370, 365, 178, 89, 1682, 85, 1850, 137, 106, 533, 2210, 100, 98, 125, 298, 185, 2810, 117, 146, 113, 370

Offset: 1

Daniel Suteu, Jan 25 2019

nonn

When n is a prime number, a(n) is greater than all the previous terms.

If n = 4*x*y, then a(n) is the smallest integer solution of the form 4*(x^2 + y^2), with rational values x and y.

			For n = 3, a(3) = 10, which is the smallest integer k such that k+2*n and k-2*n are both squares: 10+2*3 = 4^2 and 10-2*3 = 2^2.
For n=1..10, the following {a(n)-2*n, a(n)+2*n} pairs of squares are produced: {0, 4}, {1, 9}, {4, 16}, {0, 16}, {16, 36}, {1, 25}, {36, 64}, {4, 36}, {0, 36}, {9, 49}.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Difference of two squares

Cf. A000290, A029744, A033676. A033677, A056737, A061057, A063655, A087711.

f:= proc(n) local d;
d:= max(select(t -> t^2 <= n, numtheory:-divisors(n)));
d^2 + (n/d)^2
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

Array[Block[{k = 1}, While[Nand @@ Map[IntegerQ, Sqrt[k + 2 {-#, #}]], k++]; k] &, 57] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = for(k=2*n, oo, if(issquare(k+2*n) && issquare(k-2*n), return(k)));

a(n) = my(d=divisors(n)); vecmin(vector(#d, k, 4*((d[k]/2)^2 + (n/d[k]/2)^2)));

a(n^2) = 2 * n^2.
a(p) = p^2 + 1, for p prime.
a(n) = A063655(n)^2 - 2*n.
a(n) = A056737(n)^2 + 2*n.
a(n!) = A061057(n)^2 + 2*n!.
a(n) = A033676(n)^2 + A033677(n)^2. - Robert Israel, Feb 17 2019
a(n) = Min_{d|n} ((n/d)^2 + d^2). - Ridouane Oudra, Mar 17 2024

A317490 a(n) = min {i - j} where i*j is a factorization of the n-th partial product of the semiprimes (A112141).

Original entry on oeis.org

0, 2, 6, 3, 12, 3, 126, 153, 765, 1050, 5348, 14850, 85050, 501200, 91280, 3661983, 25633881, 66271296, 215467945, 254861640, 5311480020, 75327142968, 122703152000, 2187956957004, 3449084839200, 19305922856220, 11327171375520, 58038845751810, 2222926571960640

Offset: 1

Robert G. Wilson v, Jul 29 2018

nonn

			a(1) =   0 since the first semiprime is    4 =    2 *    2;
a(2) =   2 since 4*6               =      24 =    4 *    6;
a(3) =   6 since 4*6*9             =     216 =   12 *   18;
a(4) =   3 since 4*6*9*10          =    2160 =   45 *   48;
a(5) =  12 since 4*6*9*10*14       =   30240 =  168 *  180;
a(6) =   3 since 4*6*9*10*14*15    =  453600 =  172 *  175;
a(7) = 126 since 4*6*9*10*14*15*21 = 9525600 = 3024 * 3150; etc.

Inspired by A003681, and analogous to A061057 and A061060.

Cf. A112141.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; a[n_] := Block[{sp = Times @@ Array[SemiPrime@# &, n], d}, d = DivisorSigma[0, sp]/2; -Subtract @@  Take[ Divisors@ sp, {d, d + 1}]]; a[1] = 0; Array[a, 29]

cp's OEIS Frontend

A060777 Larger central (or median) divisor of n!.

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A060796 Upper central divisor of n-th primorial.

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A060795 Write product of first n primes as x*y with x

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A061060 Write product of first n primes as x*y with x

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A038667 Minimal value of |product(A) - product(B)| where A and B are a partition of {1,2,3,...,n}.

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A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n.

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Sage

A323728 a(n) is the smallest number k such that both k-2*n and k+2*n are squares.

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A323728 a(n) is the smallest number k such that both k-2n and k+2n are squares.